Editing Leech lattice (section)

==Characterization==

The Leech lattice Λ<sub>24</sub> is the unique lattice in 24-dimensional [[Euclidean space]], '''E'''<sup>24</sup>, with the following list of properties:

*It is [[unimodular lattice|unimodular]]; i.e., it can be generated by the columns of a certain 24&times;24 [[matrix (mathematics)|matrix]] with [[determinant]]&nbsp;1.
*It is even; i.e., the square of the length of each vector in Λ<sub>24</sub> is an even integer.
*The length of every non-zero vector in Λ<sub>24</sub> is at least 2.

The last condition is equivalent to the condition that unit balls centered at the points of Λ<sub>24</sub> do not overlap.  Each is tangent to 196,560 neighbors, and this is known to be the largest number of non-overlapping 24-dimensional unit balls that can [[kissing number|simultaneously touch a single unit ball]]. This arrangement of 196,560 unit balls centred about another unit ball is so efficient that there is no room to move any of the balls; this configuration, together with its mirror-image, is the ''only'' 24-dimensional arrangement where 196,560 unit balls simultaneously touch another. This property is also true in 1, 2 and 8 dimensions, with 2, 6 and 240 unit balls, respectively, based on the [[integer lattice]], [[hexagonal tiling]] and [[E8 lattice|E<sub>8</sub> lattice]], respectively.

It has no [[root system]] and in fact is the first [[unimodular lattice]] with no ''roots'' (vectors of norm less than 4), and therefore has a centre density of 1. By multiplying this value by the volume of a unit ball in 24 dimensions, <math>\tfrac{\pi^{12}}{12!}</math>, one can derive its absolute density.

{{harvtxt|Conway|1983}} showed that the Leech lattice is isometric  to the set of simple roots (or the [[Dynkin diagram]]) of the [[reflection group]] of the 26-dimensional even Lorentzian unimodular lattice [[II25,1|II<sub>25,1</sub>]]. By comparison, the Dynkin diagrams of II<sub>9,1</sub> and II<sub>17,1</sub> are finite.